Chaotic time dependence in a disordered spin system

L. R.G. Fontes; M. Isopi; C. M. Newman

doi:10.1007/s004400050244

Chaotic time dependence in a disordered spin system

L. R.G. Fontes, M. Isopi, C. M. Newman

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Stochastic Ising and voter models on ℤ^d are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t → ∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) - which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature - if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d ≥ 2. These results are based on an analysis of the "localization" properties of Random Walks with Random Rates.

Original language	English (US)
Pages (from-to)	417-443
Number of pages	27
Journal	Probability Theory and Related Fields
Volume	115
Issue number	3
DOIs	https://doi.org/10.1007/s004400050244
State	Published - Nov 1999

Keywords

Chaotic time dependence
Disordered spin system
Random environment
Random walk
Stochastic Ising model
Voter model

ASJC Scopus subject areas

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

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10.1007/s004400050244

Cite this

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AU - Isopi, M.

AU - Newman, C. M.

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N2 - Stochastic Ising and voter models on ℤd are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t → ∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) - which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature - if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d ≥ 2. These results are based on an analysis of the "localization" properties of Random Walks with Random Rates.

AB - Stochastic Ising and voter models on ℤd are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t → ∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) - which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature - if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d ≥ 2. These results are based on an analysis of the "localization" properties of Random Walks with Random Rates.

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KW - Random environment

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KW - Voter model

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Chaotic time dependence in a disordered spin system

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Keywords

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